Spherical Tensor Operators in NMR

(a) The expectation value of a component 〈α|V̂i|α〉 remains unchanged in the rotated frame (with respect to the transformed operator), i.e., 〈α′|V̂ ′ i |α′〉 = 〈α|V̂i|α〉 where |α〉 −→ D(R)|α〉 = |α′〉 and V̂ ′ i is the operator transform. The expectation value in the two bases must be the same, because we are assuming that space is isotropic, and all physical observables and physical laws must remain invariant under the rotation of the entire system, including the measuring apparatus. In such a scenario, V̂ ′ i |α′〉 = V̂i|α〉. (b) The expectation value 〈α|V̂i|α〉 is unchanged with respect to transformed operator and kets, but with respect to the kets and operators in the un-rotated frame, it transforms like the components of a cartesian vector, i.e., 〈α|V̂i|α〉 −→ ∑ j Rij〈α|V̂j |α〉 = 〈α′|V̂i|α′〉 = 〈α|D†(R)V̂iD(R)|α〉. This last equation is not to be confused with the equality 〈α′|V̂ ′ i |α′〉 = 〈α|V̂i|α〉, in which both the operator and the kets are transformed. (c) Since the matrix with elements Rij is orthogonal, i.e., ∑ k RikRjk = δij , (1c) can also be written as: